test.grid<-grid.list(5) ggplot(subset(make.spheres(test.grid),phase),aes(x,y,fill=phase))+ geom_raster(fill="gray50",alpha=0.75)+ coord_equal()+ theme_bw(20)
require(ggplot2)
require(lattice) # nicer scatter plots
require(plyr)
require(grid) # contains the arrow function
require(biOps)
require(doMC) # for parallel code
require(EBImage)
require(reshape2) # for the melt function
## To install EBImage
# source("http://bioconductor.org/biocLite.R")
# biocLite("EBImage")
require(png)
require(gridExtra)
# start parallel environment
registerDoMC()
# functions for converting images back and forth
im.to.df<-function(in.img) {
out.im<-expand.grid(x=1:nrow(in.img),y=1:ncol(in.img))
out.im$val<-as.vector(in.img)
out.im
}
df.to.im<-function(in.df,val.col="val",inv=F) {
in.vals<-in.df[[val.col]]
if(class(in.vals[1])=="logical") in.vals<-as.integer(in.vals*255)
if(inv) in.vals<-255-in.vals
out.mat<-matrix(in.vals,nrow=length(unique(in.df$x)),byrow=F)
attr(out.mat,"type")<-"grey"
out.mat
}
ddply.cutcols<-function(...,cols=1) {
# run standard ddply command
cur.table<-ddply(...)
cutlabel.fixer<-function(oVal) {
sapply(oVal,function(x) {
cnv<-as.character(x)
mean(as.numeric(strsplit(substr(cnv,2,nchar(cnv)-1),",")[[1]]))
})
}
cutname.fixer<-function(c.str) {
s.str<-strsplit(c.str,"(",fixed=T)[[1]]
t.str<-strsplit(paste(s.str[c(2:length(s.str))],collapse="("),",")[[1]]
paste(t.str[c(1:length(t.str)-1)],collapse=",")
}
for(i in c(1:cols)) {
cur.table[,i]<-cutlabel.fixer(cur.table[,i])
names(cur.table)[i]<-cutname.fixer(names(cur.table)[i])
}
cur.table
}
show.pngs.as.grid<-function(file.list,title.fun,zoom=1) {
preparePng<-function(x) rasterGrob(readPNG(x,native=T,info=T),width=unit(zoom,"npc"),interp=F)
labelPng<-function(x,title="junk") (qplot(1:300, 1:300, geom="blank",xlab=NULL,ylab=NULL,asp=1)+
annotation_custom(preparePng(x))+
labs(title=title)+theme_bw(24)+
theme(axis.text.x = element_blank(),
axis.text.y = element_blank()))
imgList<-llply(file.list,function(x) labelPng(x,title.fun(x)) )
do.call(grid.arrange,imgList)
}
## Standard image processing tools which I use for visualizing the examples in the script
commean.fun<-function(in.df) {
ddply(in.df,.(val), function(c.cell) {
weight.sum<-sum(c.cell$weight)
data.frame(xv=mean(c.cell$x),
yv=mean(c.cell$y),
xm=with(c.cell,sum(x*weight)/weight.sum),
ym=with(c.cell,sum(y*weight)/weight.sum)
)
})
}
colMeans.df<-function(x,...) as.data.frame(t(colMeans(x,...)))
pca.fun<-function(in.df) {
ddply(in.df,.(val), function(c.cell) {
c.cell.cov<-cov(c.cell[,c("x","y")])
c.cell.eigen<-eigen(c.cell.cov)
c.cell.mean<-colMeans.df(c.cell[,c("x","y")])
out.df<-cbind(c.cell.mean,
data.frame(vx=c.cell.eigen$vectors[1,],
vy=c.cell.eigen$vectors[2,],
vw=sqrt(c.cell.eigen$values),
th.off=atan2(c.cell.eigen$vectors[2,],c.cell.eigen$vectors[1,]))
)
})
}
vec.to.ellipse<-function(pca.df) {
ddply(pca.df,.(val),function(cur.pca) {
# assume there are two vectors now
create.ellipse.points(x.off=cur.pca[1,"x"],y.off=cur.pca[1,"y"],
b=sqrt(5)*cur.pca[1,"vw"],a=sqrt(5)*cur.pca[2,"vw"],
th.off=pi/2-atan2(cur.pca[1,"vy"],cur.pca[1,"vx"]),
x.cent=cur.pca[1,"x"],y.cent=cur.pca[1,"y"])
})
}
# test function for ellipse generation
# ggplot(ldply(seq(-pi,pi,length.out=100),function(th) create.ellipse.points(a=1,b=2,th.off=th,th.val=th)),aes(x=x,y=y))+geom_path()+facet_wrap(~th.val)+coord_equal()
create.ellipse.points<-function(x.off=0,y.off=0,a=1,b=NULL,th.off=0,th.max=2*pi,pts=36,...) {
if (is.null(b)) b<-a
th<-seq(0,th.max,length.out=pts)
data.frame(x=a*cos(th.off)*cos(th)+b*sin(th.off)*sin(th)+x.off,
y=-1*a*sin(th.off)*cos(th)+b*cos(th.off)*sin(th)+y.off,
id=as.factor(paste(x.off,y.off,a,b,th.off,pts,sep=":")),...)
}
deform.ellipse.draw<-function(c.box) {
create.ellipse.points(x.off=c.box$x[1],
y.off=c.box$y[1],
a=c.box$a[1],
b=c.box$b[1],
th.off=c.box$th[1],
col=c.box$col[1])
}
bbox.fun<-function(in.df) {
ddply(in.df,.(val), function(c.cell) {
c.cell.mean<-colMeans.df(c.cell[,c("x","y")])
xmn<-emin(c.cell$x)
xmx<-emax(c.cell$x)
ymn<-emin(c.cell$y)
ymx<-emax(c.cell$y)
out.df<-cbind(c.cell.mean,
data.frame(xi=c(xmn,xmn,xmx,xmx,xmn),
yi=c(ymn,ymx,ymx,ymn,ymn),
xw=xmx-xmn,
yw=ymx-ymn
))
})
}
# since the edge of the pixel is 0.5 away from the middle of the pixel
emin<-function(...) min(...)-0.5
emax<-function(...) max(...)+0.5
extents.fun<-function(in.df) {
ddply(in.df,.(val), function(c.cell) {
c.cell.mean<-colMeans.df(c.cell[,c("x","y")])
out.df<-cbind(c.cell.mean,data.frame(xmin=c(c.cell.mean$x,emin(c.cell$x)),
xmax=c(c.cell.mean$x,emax(c.cell$x)),
ymin=c(emin(c.cell$y),c.cell.mean$y),
ymax=c(emax(c.cell$y),c.cell.mean$y)))
})
}
th_fillmap.fn<-function(max.val) scale_fill_gradientn(colours=rainbow(10),limits=c(0,max.val))
author: Kevin Mader date: 17 April 2014 width: 1440 height: 900 css: ../template.css transition: rotate
ETHZ: 227-0966-00L
Comparsion of Tracking Methods in Biology
Multiple Hypothesis Testing
The course has covered imaging enough and there have been a few quantitative metrics, but "big" has not really entered.
What does big mean?
So what is "big" imaging
We can say that it looks like, but many pieces of quantitative information are difficult to extract
Understanding the flow of liquids and mixtures is important for many processes
Deformation is similarly important since it plays a significant role in the following scenarios
The first step of any of these analyses is proper experimental design. Since there is always
There are always trade-offs to be made between getting the best possible high-resolution nanoscale dynamics and capturing the system level behavior.
In many cases, experimental data is inherited and little can be done about the design, but when there is still the opportunity, simulations provide a powerful tool for tuning and balancing a large number parameters
Simulations also provide the ability to pair post-processing to the experiments and determine the limits of tracking.
Going back to our original cell image
We have at least a few samples (or different regions), large number of metrics and an almost as large number of parameters to tune
We start with a starting image
# Fill Image code
# ... is for extra columns in the data set
fill.img.fn<-function(in.img,step.size=1,...) {
xr<-range(in.img$x)
yr<-range(in.img$y)
ddply(expand.grid(x=seq(xr[1],xr[2],step.size),
y=seq(yr[1],yr[2],step.size)),
.(x,y),
function(c.pos) {
ix<-c.pos$x[1]
iy<-c.pos$y[1]
nset<-subset(in.img,x==ix & y==iy)
if(nrow(nset)<1) nset<-data.frame(x=ix,y=iy,val=0,...)
nset
})
}
make.spheres<-function(sph.list,base.gr=seq(-1,1,length.out=40)) {
start.image<-expand.grid(x=base.gr,y=base.gr)
start.image$val<-c(0)
for(i in 1:nrow(sph.list)) {
start.image$val<-with(start.image,
val + (
((x-sph.list[i,"x"])^2+(y-sph.list[i,"y"])^2)<
sph.list[i,"r"]^2)
)
}
start.image$phase<-with(start.image,ifelse(val>0,TRUE,FALSE))
start.image
}
rand.list<-function(n.pts,r=0.15,min=-1,max=1) data.frame(x=runif(n.pts,min=-1,max=1),y=runif(n.pts,min=-1,max=1),r=r)
grid.list<-function(n.pts,r=0.15,min=-1,max=1) cbind(expand.grid(x=seq(min,max,length.out=n.pts),y=seq(min,max,length.out=n.pts)),r=r)
test.grid<-grid.list(5) ggplot(subset(make.spheres(test.grid),phase),aes(x,y,fill=phase))+ geom_raster(fill="gray50",alpha=0.75)+ coord_equal()+ theme_bw(20)
\[ \vec{v}(\vec{x})=\langle 0,0.1 \rangle \]
ggplot(subset(make.spheres(test.grid),phase),aes(x,y))+
geom_raster(fill="gray50",alpha=0.75)+
geom_segment(data=cbind(test.grid,xv=0,yv=0.1),
aes(xend=x+xv,yend=y+yv),arrow=arrow(length = unit(0.3,"cm")))+
coord_equal()+
theme_bw(20)
\[ \vec{v}(\vec{x})=0.3\frac{\vec{x}}{||\vec{x}||}\times \langle 0,0,1 \rangle \]
test.grid$xyr<-with(test.grid,sqrt(x^2+y^2))
ggplot(subset(make.spheres(test.grid),phase),aes(x,y))+
geom_raster(fill="gray50",alpha=0.75)+
geom_segment(data=with(test.grid,cbind(test.grid,xv=-0.3*y/xyr,yv=0.3*x/xyr)),
aes(xend=x+xv,yend=y+yv),arrow=arrow(length = unit(0.3,"cm")))+
coord_equal()+
theme_bw(20)
\[ \vec{v}(\vec{x})=\langle 0,0.1 \rangle \]
many.frames<-ldply(seq(0,2,length.out=9),function(in.offset) {
cbind(
make.spheres(data.frame(x=test.grid$x,y=test.grid$y+in.offset,r=test.grid$r)),
offset=in.offset
)
})
ggplot(subset(many.frames,phase),aes(x,y))+
geom_raster(fill="gray50",alpha=0.75)+
coord_equal()+
facet_wrap(~offset)+
labs(title="Different Frames in Linear Flow Image")+
theme_bw(20)
\[ \vec{v}(\vec{x})=0.3\frac{\vec{x}}{||\vec{x}||}\times \langle 0,0,1 \rangle \]
many.frames<-cbind(make.spheres(test.grid),offset=0)
last.frame<-test.grid
for(in.offset in seq(0,2,length.out=9)) {
last.frame$xyr<-with(last.frame,sqrt(x^2+y^2))
last.frame$xyr[which(last.frame$xyr==0)]<-1
last.frame<-with(last.frame,data.frame(x=x-0.05*y/xyr,y=y+0.05*x/xyr,r=r))
many.frames<-rbind(many.frames,
cbind(make.spheres(last.frame),offset=in.offset)
)
}
ggplot(subset(many.frames,phase),aes(x,y))+
geom_raster(fill="gray50",alpha=0.75)+
coord_equal()+
facet_wrap(~offset)+
labs(title="Different Frames in Spiral Flow")+
theme_bw(20)
Under perfect imaging and experimental conditions objects should not appear and reappear but due to
It is common for objects to appear and vanish regularly in an experiment.
many.frames<-ldply(seq(0,1.,length.out=9),function(in.offset) {
c.grid<-test.grid[sample(nrow(test.grid), 18), ]
cbind(
make.spheres(data.frame(x=c.grid$x,y=c.grid$y+in.offset,r=c.grid$r)),
offset=in.offset
)
})
ggplot(subset(many.frames,phase),aes(x,y))+
geom_raster(fill="gray50",alpha=0.75)+
coord_equal()+
facet_wrap(~offset)+
labs(title="Different Frames in Linear Flow Image")+
theme_bw(20)
Even perfect spherical objects do not move in a straight line. The jitter can be seen as a stochastic variable with a random magnitude (\(a\)) and angle (\(b\)). This is then sampled at every point in the field
\[ \vec{v}(\vec{x})=\vec{v}_L(\vec{x})+||a||\measuredangle b \]
ggplot(subset(make.spheres(test.grid),phase),aes(x,y))+
geom_raster(fill="gray50",alpha=0.75)+
geom_segment(data=cbind(test.grid,
xv=0+runif(nrow(test.grid),min=-0.05,max=0.05),
yv=0.1+runif(nrow(test.grid),min=-0.05,max=0.05)),
aes(xend=x+xv,yend=y+yv),arrow=arrow(length = unit(0.3,"cm")))+
coord_equal()+
theme_bw(20)
Over many frames this can change the path significantly
last.frame<-test.grid[,c("x","y","r")]
last.frame$id<-1:nrow(last.frame)
many.frames<-cbind(make.spheres(last.frame),offset=0)
many.grids<-cbind(last.frame,offset=0)
for(in.offset in cumsum(rep(0.2,8))) {
last.frame<-with(last.frame,
data.frame(x=x+runif(nrow(last.frame),min=-0.1,max=0.1),
y=y+0.2+runif(nrow(last.frame),min=-0.1,max=0.1),
r=r)
)
last.frame$id<-1:nrow(last.frame)
many.frames<-rbind(many.frames,
cbind(make.spheres(last.frame),offset=in.offset)
)
many.grids<-rbind(many.grids,
cbind(last.frame,offset=in.offset)
)
}
ggplot(subset(many.frames,phase),aes(x,y))+
geom_raster(fill="gray50",alpha=0.75)+
coord_equal()+
facet_wrap(~offset)+
labs(title="Different Frames in Linear Flow Image")+
theme_bw(20)
The simulation can be represented in a more clear fashion by using single lines to represent each spheroid
ggplot(many.grids,aes(x,y,))+ geom_path(aes(color=id,group=id))+ coord_equal()+ labs(title="Different Paths in Linear Jittered Flow Image")+ scale_color_gradientn(colours=rainbow(10))+ theme_bw(20)
We see that visually tracking samples can be difficult and there are a number of parameters which affect the ability for us to clearly see the tracking.
We thus try to quantify the limits of these parameters for different tracking methods in order to design experiments better.
While there exist a number of different methods and complicated approaches for tracking, for experimental design it is best to start with the simplist, easiest understood method. The limits of this can be found and components added as needed until it is possible to realize the experiment
We then return to nearest neighbor which means we track a point (\(\vec{P}_0\)) from an image (\(I_0\)) at \(t_0\) to a point (\(\vec{P}_1\)) in image (\(I_1\)) at \(t_1\) by
\[ \vec{P}_1=\textrm{argmin}(||\vec{P}_0-\vec{y}|| \forall \vec{y}\in I_1) \]
In the following examples we will use simple metrics for scoring fits where the objects are matched and the number of misses is counted.
There are a number of more sensitive scoring metrics which can be used, by finding the best submatch for a given particle since the number of matches and particles does not always correspond. See the papers at the beginning for more information
source('~/Dropbox/TIPL/src/R/trackingCode.R')
source('~/Dropbox/WorkRelated/TrackTopology/simFlow.R')
Input flow from simulation
\[ \vec{v}(\vec{x})=\langle 0,0,0.05 \rangle+||0.01||\measuredangle b \]
# Generate a simple synthetic dataset in.frames<-generate.frames(base.rand=0.01,crop.size=c(-10,10),n.objects=20,n.frames=10,flow.rate=0.05,force.2d=T) all.tracks<-track.frames(in.frames,offset=c(0,0,0.1),run.offset=T,run.adaptive=T,maxVolPenalty=NA)
ggplot(do.call(rbind,in.frames),
aes(x=POS_X,y=POS_Z,color=as.factor(REAL_LACUNA_NUMBER)))+
geom_path(aes(linetype="Original"))+
labs(x="X Position",y="Z Position",color="Object ID",title="Flow Simulation Results")+
theme_bw(20)
Nearest Neighbor Tracking
ggplot(do.call(rbind,in.frames),
aes(x=POS_X,y=POS_Z,color=as.factor(REAL_LACUNA_NUMBER)))+
#geom_path(aes(linetype="Original"))+
geom_path(data=subset(all.tracks,Matching=="No Offset"),aes(linetype="Tracked"),size=2,alpha=0.5)+
facet_wrap(~Matching)+
labs(x="X Position",y="Z Position",color="Object ID",title="Tracking Results")+
theme_bw(20)
Input flow from simulation
\[ \vec{v}(\vec{x})=\langle 0,0,0.01 \rangle+||0.05||\measuredangle b \]
# Generate a simple synthetic dataset
in.frames<-generate.frames(base.rand=0.1,crop.size=c(-10,10),n.objects=20,n.frames=20,flow.rate=0.01,
force.2d=T,rand.fun=function(x,min=0,max=1) rnorm(x,(min+max)/2,(max-min)/2))
all.tracks<-track.frames(in.frames,offset=c(0,0,0.1),run.offset=T,run.adaptive=T,maxVolPenalty=NA)
ggplot(do.call(rbind,in.frames),
aes(x=POS_X,y=POS_Z,color=as.factor(REAL_LACUNA_NUMBER)))+
geom_path(aes(linetype="Original"))+
labs(x="X Position",y="Z Position",color="Object ID",title="Flow Simulation Results")+
theme_bw(20)
Nearest Neighbor Tracking
ggplot(do.call(rbind,in.frames),
aes(x=POS_X,y=POS_Z,color=as.factor(REAL_LACUNA_NUMBER)))+
geom_path(aes(linetype="Original"))+
geom_path(data=subset(all.tracks,Matching=="No Offset"),aes(linetype="Tracked"),size=2,alpha=0.5)+
facet_wrap(~Matching)+
labs(x="X Position",y="Z Position",color="Object ID",title="Tracking Results")+
theme_bw(20)
We can then quantify the success rate of each algorithm on the data set using the very simple match and mismatch metrics
c.tracks<-subset(all.tracks,Matching=="No Offset")
c.tracks<-all.tracks
c.subtrack<-subset(c.tracks,abs(D_REAL_LACUNA_NUMBER)>0)
ggplot(do.call(rbind,in.frames),
aes(x=POS_X,y=POS_Z))+
geom_path(aes(linetype="Original",color=as.factor(REAL_LACUNA_NUMBER)))+
geom_path(data=c.tracks,aes(linetype="Tracked",color=as.factor(REAL_LACUNA_NUMBER)),size=2,alpha=0.5)+
geom_point(data=c.subtrack,aes(shape="Missed"),size=5,alpha=0.5,color="red")+
facet_wrap(~Matching)+
labs(x="X Position",y="Z Position",color="Object ID",title="Tracking Results",fill="Misses")+
theme_bw(20)
c.tracks<-subset(all.tracks,Matching=="No Offset")
c.tracks<-all.tracks
c.subtrack<-subset(c.tracks,abs(D_REAL_LACUNA_NUMBER)>0)
ggplot(do.call(rbind,in.frames),
aes(x=POS_X,y=POS_Z))+
stat_binhex(data=c.subtrack,bins=5,drop=F,alpha=1)+
scale_fill_gradient2(low="white",high="red")+
facet_wrap(~Matching)+
labs(x="X Position",y="Z Position",color="Object ID",title="Tracking Results",fill="Misses")+
theme_bw(20)
pout<-function(x) paste(round(1000*(1-mean(x)))/10,"%",sep="")
match.table<-ddply.cutcols(all.tracks,.(cut_interval(sample,10)),function(all.sample) {
data.frame(
BIJ_MATCHA=with(subset(all.sample,Matching=="No Offset"),pout(BIJ_MATCH)),
BIJ_MATCHB=with(subset(all.sample,Matching=="Fix Offset"),pout(BIJ_MATCH)),
BIJ_MATCHC=with(subset(all.sample,Matching=="Adaptive Offset"),pout(BIJ_MATCH))
)
})
names(match.table)<-c("Time","NN","ONN","ANN")
kable(match.table)
| Time | NN | ONN | ANN |
|---|---|---|---|
| 1.9 | 22.5% | 22.5% | 25% |
| 3.7 | 22.5% | 25% | 25% |
| 5.5 | 32.5% | 35% | 32.5% |
| 7.3 | 32.5% | 32.5% | 27.5% |
| 9.1 | 20% | 30% | 22.5% |
| 10.9 | 20% | 20% | 25% |
| 12.7 | 12.5% | 17.5% | 17.5% |
| 14.5 | 17.5% | 25% | 20% |
| 16.3 | 15% | 17.5% | 12.5% |
| 18.1 | 22.5% | 27.5% | 25% |
match.qual<-function(in.tracks) ddply(in.tracks,.(Matching),function(c.subset) {
data.frame(Obj.Matched=sum(c.subset$D_REAL_LACUNA_NUMBER==0),
Obj.Missed=sum(abs(c.subset$D_REAL_LACUNA_NUMBER)>0))
})
rand.fun.norm<-function(n,minv,maxv) rnorm(n,(maxv+minv)/2,(maxv-minv)/2)
n.iters<-20
z.vel<-0.1
n.frames<-2
jd.gen.fun<-function(c.jitter,c.obj.count,...) {
test.data<-generate.frames(base.rand=c.jitter*z.vel,crop.size=c(-10,10),n.objects=c.obj.count,n.frames=n.frames,rand.fun=rand.fun.norm,...)
test.tracks<-track.frames(test.data,offset=c(0,0,z.vel),run.offset=T,run.adaptive=T,maxVolPenalty=NA)
cbind(jitter=c.jitter,obj.count=c.obj.count,
mean_obj_spacing=((1/c.obj.count)^0.33)/z.vel,
match.qual(test.tracks))
}
jitter.vals<-rep(seq(0,2.5,length.out=15),n.iters)
jitter.full<-
jitter.summary.3d<-ddply(ldply(jitter.vals,function(c.jitter) jd.gen.fun(c.jitter,20),.parallel=T),
.(jitter,Matching),
function(c.subset) {
data.frame(Matched=100*sum(c.subset$Obj.Matched)/(sum(c.subset$Obj.Missed)+sum(c.subset$Obj.Matched)))
})
jitter.summary.2d<-ddply(ldply(jitter.vals,function(c.jitter) jd.gen.fun(c.jitter,20,force.2d=T),.parallel=T),
.(jitter,Matching),
function(c.subset) {
data.frame(Matched=100*sum(c.subset$Obj.Matched)/(sum(c.subset$Obj.Missed)+sum(c.subset$Obj.Matched)))
})
ggplot(rbind(cbind(jitter.summary.3d,geom="3D"),cbind(jitter.summary.2d,geom="2D")),
aes(x=100*jitter,y=Matched,color=geom))+
geom_line()+geom_point()+facet_wrap(~Matching)+
theme_bw(24)+labs(x="Position Jitter (% of Velocity)",y="% of Bubbles Matched",color="Matching Type")
n.iters<-20
registerDoMC(8) # divide the jobs better
jitter.vals<-seq(0,2,length.out=4)
irseq<-function(a,b,length.out) {1/seq(1/b^(1/3),1/a^(1/3),length.out=length.out)^3} # seq for inverted numbers
obj.count<-irseq(25,2500,length.out=3)
jit.bub<-merge(obj.count,jitter.vals)
jd.vals<-mapply(list,rep(jit.bub[,1],n.iters),rep(jit.bub[,2],n.iters),SIMPLIFY=F)
jd.full<-ldply(jd.vals,.parallel=T,
function(c.in) jd.gen.fun(c.in[[2]],c.in[[1]]))
jd.summary<-ddply(jd.full,.(jitter,mean_obj_spacing,Matching),function(c.subset) {
data.frame(obj.count=c.subset$obj.count[1],
Matched=100*sum(c.subset$Obj.Matched)/(sum(c.subset$Obj.Missed)+sum(c.subset$Obj.Matched)),
obj.matched=sum(c.subset$Obj.Matched),
obj.found=100*with(c.subset,sum(Obj.Matched)/(n.iters*(n.frames-1)*obj.count[1])))
})
ggplot(jd.summary,aes(x=100*jitter,y=Matched,color=as.factor(round(100*mean_obj_spacing))))+ geom_line()+geom_point()+facet_wrap(~Matching)+ theme_bw(24)+labs(x="Position Jitter (% of Velocity)",y="% of Obj Matched",color="Obj.Spacing\n(% of Velocity)")
ggplot(jd.summary,aes(x=100*jitter,y=100*mean_obj_spacing,fill=Matched))+ geom_tile()+facet_wrap(~Matching)+ labs(x="Position Jitter (% of Velocity)",fill="% of Obj Matched",y="Obj.Spacing (% of Velocity)")+ theme_bw(24)
kable(jd.summary)
| jitter | mean_obj_spacing | Matching | obj.count | Matched | obj.matched | obj.found |
|---|---|---|---|---|---|---|
| 0.0000 | 0.7563 | No Offset | 2500.0 | 2.156 | 1078 | 2.156 |
| 0.0000 | 0.7563 | Fix Offset | 2500.0 | 100.000 | 50000 | 100.000 |
| 0.0000 | 0.7563 | Adaptive Offset | 2500.0 | 100.000 | 50000 | 100.000 |
| 0.0000 | 2.1113 | No Offset | 111.4 | 41.622 | 924 | 41.478 |
| 0.0000 | 2.1113 | Fix Offset | 111.4 | 100.000 | 2220 | 99.655 |
| 0.0000 | 2.1113 | Adaptive Offset | 111.4 | 100.000 | 2220 | 99.655 |
| 0.0000 | 3.4568 | No Offset | 25.0 | 80.600 | 403 | 80.600 |
| 0.0000 | 3.4568 | Fix Offset | 25.0 | 100.000 | 500 | 100.000 |
| 0.0000 | 3.4568 | Adaptive Offset | 25.0 | 100.000 | 500 | 100.000 |
| 0.6667 | 0.7563 | No Offset | 2500.0 | 3.922 | 1961 | 3.922 |
| 0.6667 | 0.7563 | Fix Offset | 2500.0 | 8.122 | 4061 | 8.122 |
| 0.6667 | 0.7563 | Adaptive Offset | 2500.0 | 8.132 | 4066 | 8.132 |
| 0.6667 | 2.1113 | No Offset | 111.4 | 37.703 | 837 | 37.572 |
| 0.6667 | 2.1113 | Fix Offset | 111.4 | 54.144 | 1202 | 53.957 |
| 0.6667 | 2.1113 | Adaptive Offset | 111.4 | 53.919 | 1197 | 53.733 |
| 0.6667 | 3.4568 | No Offset | 25.0 | 69.000 | 345 | 69.000 |
| 0.6667 | 3.4568 | Fix Offset | 25.0 | 81.600 | 408 | 81.600 |
| 0.6667 | 3.4568 | Adaptive Offset | 25.0 | 80.000 | 400 | 80.000 |
| 1.3333 | 0.7563 | No Offset | 2500.0 | 2.326 | 1163 | 2.326 |
| 1.3333 | 0.7563 | Fix Offset | 2500.0 | 2.580 | 1290 | 2.580 |
| 1.3333 | 0.7563 | Adaptive Offset | 2500.0 | 2.574 | 1287 | 2.574 |
| 1.3333 | 2.1113 | No Offset | 111.4 | 22.748 | 505 | 22.669 |
| 1.3333 | 2.1113 | Fix Offset | 111.4 | 25.541 | 567 | 25.452 |
| 1.3333 | 2.1113 | Adaptive Offset | 111.4 | 25.586 | 568 | 25.497 |
| 1.3333 | 3.4568 | No Offset | 25.0 | 51.600 | 258 | 51.600 |
| 1.3333 | 3.4568 | Fix Offset | 25.0 | 57.600 | 288 | 57.600 |
| 1.3333 | 3.4568 | Adaptive Offset | 25.0 | 57.400 | 287 | 57.400 |
| 2.0000 | 0.7563 | No Offset | 2500.0 | 1.010 | 505 | 1.010 |
| 2.0000 | 0.7563 | Fix Offset | 2500.0 | 1.168 | 584 | 1.168 |
| 2.0000 | 0.7563 | Adaptive Offset | 2500.0 | 1.166 | 583 | 1.166 |
| 2.0000 | 2.1113 | No Offset | 111.4 | 12.928 | 287 | 12.883 |
| 2.0000 | 2.1113 | Fix Offset | 111.4 | 13.514 | 300 | 13.467 |
| 2.0000 | 2.1113 | Adaptive Offset | 111.4 | 13.378 | 297 | 13.332 |
| 2.0000 | 3.4568 | No Offset | 25.0 | 35.000 | 175 | 35.000 |
| 2.0000 | 3.4568 | Fix Offset | 25.0 | 34.200 | 171 | 34.200 |
| 2.0000 | 3.4568 | Adaptive Offset | 25.0 | 33.400 | 167 | 33.400 |
\[ \vec{P}_f=\textrm{argmin}(||\vec{P}_0-\vec{y} || \forall \vec{y}\in I_1) \] \[ \vec{P}_i=\textrm{argmin}(||\vec{P}_f-\vec{y} || \forall \vec{y}\in I_0) \]
\[ \vec{P}_i \stackrel{?}{=} \vec{P}_0 \]
\[ \vec{P}_1=\begin{cases} ||\vec{P}_0-\vec{y} ||<\textrm{MAXD}, & \textrm{argmin}(||\vec{P}_0-\vec{y} || \forall \vec{y}\in I_1) \\ \textrm{Otherwise}, & \emptyset \end{cases}\]
\[ \vec{P}_1=\textrm{argmin}(||\vec{P}_0+\vec{v}_{offset}-\vec{y} || \forall \vec{y}\in I_1) \]
Can then be calculated in an iterative fashion where the offset is the average from all of the \(\vec{P}_1-\vec{P}_0\) vectors. It can also be performed \[ \vec{P}_1=\textrm{argmin}(||\vec{P}_0+\vec{v}_{offset}-\vec{y} || \forall \vec{y}\in I_1) \]
While nearest neighbor provides a useful starting tool it is not sufficient for truly complicated flows and datasets.
For voxel-based approachs the most common analyses are digital image correlation (or for 3D images digital volume correlation), where the correlation is calculated between two images or volumes.
Given images \(I_0(\vec{x})\) and \(I_1(\vec{x})\) at time \(t_0\) and \(t_1\) respectively. The correlation between these two images can be calculated
\[ C_{I_0,I_1}(\vec{r})=\langle I_0(\vec{x}) I_1(\vec{x}+\vec{r}) \rangle \]
# so everything is on an integer lattice (interpolation makes everything messier)
fix.grid<-function(in.grid) {
cols.nopos<-!(names(in.grid) %in% c("x","y"))
out.grid<-in.grid[,cols.nopos]
out.grid$x<-as.numeric(as.factor(in.grid$x))
out.grid$y<-as.numeric(as.factor(in.grid$y))
out.grid
}
start.grid<-grid.list(5)
start.img<-fix.grid(make.spheres(start.grid,base.gr=seq(-1,1,length.out=40)))
final.grid<-data.frame(x=with(start.grid,x),y=with(start.grid,y+0.1),r=start.grid$r)
final.img<-fix.grid(make.spheres(final.grid,base.gr=seq(-1,1,length.out=40)))
ggplot()+
geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+
geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+
coord_equal()+
labs(fill="time")+
theme_bw(20)
#' Calculate the cross correlation
#' @author Kevin Mader (kevin.mader@gmail.com)
#' Generates flow with given object count, frame count and randomness
#' the box and crop are introduced to allow for objects entering and
#' leaving the field of view
#'
#' @param img.a is the starting or I_0 image
#' @param img.b is the destination or I_1 image
#' @param tr.x is the function transforming the x coordinate
#' @param
#'
cc.imfun<-function(img.a,img.b,tr.x=function(x,y) x,tr.y=function(x,y) y) {
# get the positions in image a
x.vals<-unique(img.a$x)
y.vals<-unique(img.a$y)
# transform image b
tr.img.b<-img.b
# round is used to put everything back on an integer lattice
tr.img.b$x<-round(with(img.b,tr.x(x,y)))
tr.img.b$y<-round(with(img.b,tr.y(x,y)))
# count the overlapping pixels in the window to normalize
tr.img.b<-subset(tr.img.b,
((x %in% x.vals) & (y %in% y.vals))
)
norm.f<-nrow(tr.img.b)
if(norm.f<1) norm.f<-1
# keep only the in phase objects
tr.img.a<-subset(img.a,phase)
tr.img.b<-subset(tr.img.b,phase)
if (nrow(tr.img.a)>0 & nrow(tr.img.b)>0) {
matches<-ddply(rbind(cbind(tr.img.a,label="A"),cbind(tr.img.b,label="B")),.(x,y),
function(c.pos) {
if(nrow(c.pos)>1) data.frame(e.val=1)
else data.frame(e.val=c())
})
data.frame(e.val=sum(matches$e.val)/norm.f,count=nrow(matches),size=norm.f)
} else {
data.frame(e.val=0,count=0,size=norm.f)
}
}
cc.points<-expand.grid(vx=seq(-8,8,1),vy=seq(-8,8,1))
cc.img<-ddply(cc.points,.(vx,vy),function(c.pt) {
tr.x<-function(x,y) (x+c.pt[1,"vx"])
tr.y<-function(x,y) (y+c.pt[1,"vy"])
cc.imfun(start.img,final.img,tr.x=tr.x,tr.y=tr.y)
},.parallel=T)
ggplot(cc.img,aes(vx,vy,fill=e.val))+ geom_raster()+geom_density2d(data=subset(cc.img,e.val>0),aes(weight=e.val))+ labs(x="u",y="v",fill="Correlation",title="Correlation vs R")+ scale_fill_gradient2(high="red")+ theme_bw(25)
With highly structured / periodic samples identfying a best correlation is difficult since there are multiple maxima.
start.grid<-rand.list(12) start.img<-fix.grid(make.spheres(start.grid,base.gr=seq(-1,1,length.out=30))) final.grid<-data.frame(x=with(start.grid,x),y=with(start.grid,y+0.2),r=start.grid$r) final.img<-fix.grid(make.spheres(final.grid,base.gr=seq(-1,1,length.out=30))) ggplot()+ geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+ geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+ coord_equal()+ labs(fill="time")+ theme_bw(20)
cc.points<-expand.grid(vx=seq(-8,8,1),vy=seq(-8,8,1))
cc.img<-ddply(cc.points,.(vx,vy),function(c.pt) {
tr.x<-function(x,y) (x+c.pt[1,"vx"])
tr.y<-function(x,y) (y+c.pt[1,"vy"])
cc.imfun(start.img,final.img,tr.x=tr.x,tr.y=tr.y)
},.parallel=T)
ggplot(cc.img,aes(vx,vy,fill=e.val))+ geom_raster()+geom_density2d(data=subset(cc.img,e.val>0),aes(weight=e.val^2))+ labs(x="u",y="v",fill="Correlation",title="Correlation vs r")+ scale_fill_gradient2(high="red")+ theme_bw(25)
The correlation function can be extended by adding rotation and scaling terms to the offset making the tool more flexible but also more computationally expensive for large search spaces.
\[ C_{I_0,I_1}(\vec{r},s,\theta)= \] \[ \langle I_0(\vec{x}) I_1( \begin{bmatrix} s\cos\theta & -s\sin\theta\\ s\sin\theta & s\cos\theta \end{bmatrix} \vec{x}+\vec{r}) \rangle \]
start.grid<-rand.list(15)
rot.th<--pi/4
start.img<-fix.grid(make.spheres(start.grid,base.gr=seq(-1,1,length.out=60)))
final.grid<-data.frame(x=0.9*with(start.grid,cos(rot.th)*x-sin(rot.th)*y),
y=0.9*with(start.grid,cos(rot.th)*y+sin(rot.th)*x),r=start.grid$r)
final.img<-fix.grid(make.spheres(final.grid,base.gr=seq(-1,1,length.out=60)))
ggplot()+
geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+
geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+
coord_equal()+
labs(fill="time")+
theme_bw(20)
cc.points<-expand.grid(theta=seq(0,pi/2,length.out=8),a=seq(0.7,1,length.out=5))
mid.pt<-c(20,20)
cc.img<-ddply(cc.points,.(theta,a),function(c.pt) {
tr.x<-function(x,y)
(c.pt[1,"a"]*(cos(c.pt[1,"theta"])*(x-mid.pt[1])-sin(c.pt[1,"theta"])*(y-mid.pt[2]))+mid.pt[1])
tr.y<-function(x,y)
(c.pt[1,"a"]*(sin(c.pt[1,"theta"])*(x-mid.pt[1])+cos(c.pt[1,"theta"])*(y-mid.pt[2]))+mid.pt[1])
cc.imfun(start.img,final.img,tr.x=tr.x,tr.y=tr.y)
},.parallel=T)
ggplot(cc.img,aes(theta*180/pi,a*100,fill=e.val))+ geom_raster()+geom_density2d(data=subset(cc.img,e.val>0),aes(weight=e.val))+ labs(x="Rotation (deg)",y="Scale (%)",fill="Correlation",title="Correlation vs R")+ scale_fill_gradient2(high="red")+ theme_bw(25)
start.grid<-rand.list(20) start.img<-fix.grid(make.spheres(start.grid,base.gr=seq(-1,1,length.out=30))) final.grid<-data.frame(x=with(start.grid,sign(x)*abs(x)^1.5),y=with(start.grid,sign(y)*abs(y)^1.5),r=start.grid$r) final.img<-fix.grid(make.spheres(final.grid,base.gr=seq(-1,1,length.out=30))) ggplot()+ geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+ geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+ coord_equal()+ labs(fill="time")+ theme_bw(20)
cc.points<-expand.grid(vx=seq(-6,6,1),vy=seq(-6,6,1))
cc.img<-ddply(cc.points,.(vx,vy),function(c.pt) {
tr.x<-function(x,y) (x+c.pt[1,"vx"])
tr.y<-function(x,y) (y+c.pt[1,"vy"])
cc.imfun(start.img,final.img,tr.x=tr.x,tr.y=tr.y)
},.parallel=T)
ggplot(subset(cc.img,count>0),aes(vx,vy,fill=e.val))+ geom_raster()+geom_density2d(data=subset(cc.img,e.val>0),aes(weight=e.val^2))+ labs(x="u",y="v",fill="Correlation",title="Correlation vs r")+ scale_fill_gradient2(high="red")+ theme_bw(25)
We can approach the problem by subdividing the data into smaller blocks and then apply the digital volume correlation independently to each block.
blockify.img<-function(in.img,nx=5,ny=5,block.fn=function(c.block) c.block,...) {
out.img<-ddply(in.img,.(cut_interval(x,nx),cut_interval(y,ny)),block.fn,...)
names(out.img)[c(1:2)]<-c("x.block","y.block")
out.img$label.block<-with(out.img,paste(x.block,y.block,sep=","))
cutlabel.fixer<-function(oVal) {
sapply(oVal,function(x) {
cnv<-as.character(x)
mean(as.numeric(strsplit(substr(cnv,2,nchar(cnv)-1),",")[[1]]))
})
}
out.img$x.center<-cutlabel.fixer(out.img$x.block)
out.img$y.center<-cutlabel.fixer(out.img$y.block)
out.img
}
start.img.blocks<-blockify.img(start.img,4,4)
final.img.blocks<-blockify.img(final.img,4,4)
comb.img.blocks<-rbind(cbind(start.img.blocks,time=0),
cbind(final.img.blocks,time=1))
ggplot(subset(comb.img.blocks,phase),aes(x,y,fill=as.factor(time)))+
geom_raster(alpha=0.75)+
facet_grid(y.block~x.block,scales="free",as.table=F)+
theme_bw(10)
#' Calculate the cross correlation
#' @author Kevin Mader (kevin.mader@gmail.com)
#' Generates flow with given object count, frame count and randomness
#' the box and crop are introduced to allow for objects entering and
#' leaving the field of view
#'
#' @param bulk.img the image passed from the blockify command
#' @param f.img is the template image to compare against (since we want the whole image not just a region)
#' @param nsize is the size of the region to search
#' @param nstep is the step to use
block.corr.fun<-function(bulk.img,s.img,nsize=6,nstep=2) {
cc.points<-expand.grid(vx=seq(-nsize,nsize,nstep),vy=seq(-nsize,nsize,nstep))
ddply(cc.points,.(vx,vy),function(c.pt) {
tr.x<-function(x,y) (x+c.pt[1,"vx"])
tr.y<-function(x,y) (y+c.pt[1,"vy"])
cc.imfun(s.img,bulk.img,tr.x=tr.x,tr.y=tr.y)
})
}
w.bcf<-function(s.img,nsize=7,nstep=2) function(cur.block) block.corr.fun(cur.block,s.img,nsize=nsize,nstep=nstep)
max.corr.val<-function(in.blocks) ddply(in.blocks,.(x.center,y.center),function(in.block) {
subset(in.block,e.val>=max(in.block$e.val) & count>0)
})
corr.img.blocks<-blockify.img(final.img,4,4,block.fn=w.bcf(start.img),.parallel=T) ggplot(corr.img.blocks,aes(vx,vy,fill=e.val))+ geom_raster()+geom_density2d(data=subset(corr.img.blocks,e.val>0),aes(weight=e.val^2))+ coord_equal()+facet_grid(y.block~x.block,as.table=F)+ scale_fill_gradient2(high="red")+ labs(fill="Correlation")+ theme_bw(10)
max.block.vals<-max.corr.val(corr.img.blocks)
ggplot()+
geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+
geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+
geom_point(data=max.block.vals,aes(x.center,y.center),color="green",size=2)+
geom_segment(data=max.block.vals,aes(x.center,y.center,xend=x.center+vx,yend=y.center+vy),
arrow=arrow(length = unit(0.3,"cm")))+
coord_equal()+
labs(fill="time")+
theme_bw(20)
start.img.blocks<-blockify.img(start.img,2,2) final.img.blocks<-blockify.img(final.img,2,2) ggplot()+ geom_raster(data=subset(start.img.blocks,phase),aes(x,y,fill="0"),alpha=0.75)+ geom_raster(data=subset(final.img.blocks,phase),aes(x,y,fill="1"),alpha=0.75)+ facet_grid(y.block~x.block,scales="free",as.table=F)+ labs(fill="time")+ theme_bw(10)
corr.img.blocks<-blockify.img(final.img,2,2,block.fn=w.bcf(start.img),.parallel=T) ggplot(corr.img.blocks,aes(vx,vy,fill=e.val))+ geom_raster()+geom_density2d(data=subset(corr.img.blocks,e.val>0),aes(weight=e.val^2))+ coord_equal()+facet_grid(y.block~x.block,as.table=F)+ scale_fill_gradient2(high="red")+ labs(fill="Correlation")+ theme_bw(10)
max.block.vals<-max.corr.val(corr.img.blocks)
ggplot()+
geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+
geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+
geom_point(data=max.block.vals,aes(x.center,y.center),color="green",size=2)+
geom_segment(data=max.block.vals,aes(x.center,y.center,xend=x.center+vx,yend=y.center+vy),
arrow=arrow(length = unit(0.3,"cm")))+
coord_equal()+
labs(fill="time")+
theme_bw(20)
start.img.blocks<-blockify.img(start.img,8,8) final.img.blocks<-blockify.img(final.img,8,8) ggplot()+ geom_raster(data=subset(start.img.blocks,phase),aes(x,y,fill="0"),alpha=0.75)+ geom_raster(data=subset(final.img.blocks,phase),aes(x,y,fill="1"),alpha=0.75)+ facet_grid(y.block~x.block,scales="free",as.table=F)+ labs(fill="time")+ theme_bw(10)
corr.img.blocks<-blockify.img(final.img,8,8,block.fn=w.bcf(start.img,4),.parallel=T) ggplot(corr.img.blocks,aes(vx,vy,fill=e.val))+ geom_raster()+geom_density2d(data=subset(corr.img.blocks,e.val>0),aes(weight=e.val^2))+ coord_equal()+facet_grid(y.block~x.block,as.table=F)+ scale_fill_gradient2(high="red")+ labs(fill="Correlation")+ theme_bw(10)
max.block.vals<-max.corr.val(corr.img.blocks)
ggplot()+
geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+
geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+
geom_point(data=max.block.vals,aes(x.center,y.center),color="green",size=2)+
geom_segment(data=max.block.vals,aes(x.center,y.center,xend=x.center+vx,yend=y.center+vy),
arrow=arrow(length = unit(0.3,"cm")))+
coord_equal()+
labs(fill="time")+
theme_bw(20)
start.grid<-rand.list(25) start.img<-fix.grid(make.spheres(start.grid,base.gr=seq(-1,1,length.out=30))) final.grid<-data.frame(x=with(start.grid,x+y/5),y=with(start.grid,y),r=start.grid$r) final.img<-fix.grid(make.spheres(final.grid,base.gr=seq(-1,1,length.out=30))) ggplot()+ geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+ geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+ coord_equal()+ labs(fill="time")+ theme_bw(20)
corr.img.blocks<-blockify.img(final.img,5,5,block.fn=w.bcf(start.img,4,2),.parallel=T)
max.block.vals<-max.corr.val(corr.img.blocks)
ggplot()+
geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+
geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+
geom_point(data=max.block.vals,aes(x.center,y.center),color="green",size=2)+
geom_segment(data=max.block.vals,aes(x.center,y.center,xend=x.center+vx,yend=y.center+vy),
arrow=arrow(length = unit(0.3,"cm")))+
coord_equal()+
labs(fill="time")+
theme_bw(20)
start.grid<-rand.list(40) start.img<-fix.grid(make.spheres(start.grid,base.gr=seq(-1,1,length.out=40))) final.grid<-data.frame(x=with(start.grid,x),y=with(start.grid,y),r=start.grid$r/2) final.img<-fix.grid(make.spheres(final.grid,base.gr=seq(-1,1,length.out=40))) ggplot()+ geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+ geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+ coord_equal()+ labs(fill="time")+ theme_bw(20)
corr.img.blocks<-blockify.img(final.img,10,10,block.fn=w.bcf(start.img,4),.parallel=T)
max.block.vals<-max.corr.val(corr.img.blocks)
ggplot()+
geom_raster(data=subset(start.img,phase),aes(x,y,fill="0"),alpha=0.75)+
geom_raster(data=subset(final.img,phase),aes(x,y,fill="1"),alpha=0.75)+
geom_point(data=max.block.vals,aes(x.center,y.center),color="green",size=2)+
geom_segment(data=max.block.vals,aes(x.center,y.center,xend=x.center+vx,yend=y.center+vy),
arrow=arrow(length = unit(0.3,"cm")))+
coord_equal()+
labs(fill="time")+
theme_bw(20)
As we covered before distribution metrics like the distribution tensor can be used for tracking changes inside a sample. Of these the most relevant is the texture tensor from cellular materials and liquid foam. The texture tensor is the same as the distribution tensor except that the edges (or faces) represent physically connected / touching objects rather than touching Voronoi faces (or conversely Delaunay triangles).
These metrics can also be used for tracking the behavior of a system without tracking the single points since most deformations of a system also deform the distribution tensor and can thus be extracted by comparing the distribution tensor at different time steps.
We can take any of these approaches and quantify the deformation using a tool called the strain tensor. Strain is defined in mechanics for the simple 1D case as the change in the length against the change in the original length. \[ e = \frac{\Delta L}{L} \] While this defines the 1D case well, it is difficult to apply such metrics to voxel, shape, and tensor data.
There are a number of different ways to calculate strain and the strain tensor, but the most applicable for general image based applications is called the infinitesimal strain tensor, because the element matches well to square pixels and cubic voxels.
b.sq.ele<-0.4*data.frame(dx=c(-1,1,1,-1,-1),dy=c(-1,-1,1,1,-1))
s.grid<-expand.grid(x=c(0:3),y=c(0:3))
sq.eles<-ddply(cbind(s.grid,id=1:nrow(s.grid)),.(x,y),function(in.pt) {
data.frame(x=in.pt[1,"x"]+b.sq.ele$dx,
y=in.pt[1,"y"]+b.sq.ele$dy,
id=in.pt[1,"id"])
})
ggplot(sq.eles,aes(x,y))+
geom_polygon(aes(group=id))+
coord_equal()+
theme_bw(25)
A given strain can then be applied and we can quantify the effects by examining the change in the small element.
s.grid<-expand.grid(x=c(0:3),y=c(0:3))
rh.eles<-ddply(cbind(s.grid,id=1:nrow(s.grid)),.(x,y),function(in.pt) {
out.df<-data.frame(x=in.pt[1,"x"]+b.sq.ele$dx/(in.pt[1,"x"]+1),
y=in.pt[1,"y"]+b.sq.ele$dy/(in.pt[1,"y"]+1),
id=in.pt[1,"id"],
x.rel.sz=1,
y.rel.sz=1,
x.sz=1/(in.pt[1,"y"]+1),
y.sz=1/(in.pt[1,"x"]+1))
out.df$e11<-with(out.df,(x.sz-x.rel.sz)/x.rel.sz)
out.df$e22<-with(out.df,(y.sz-y.rel.sz)/y.rel.sz)
out.df$e12<-0.5*with(out.df,(x.sz-x.rel.sz)/y.rel.sz+(y.sz-y.rel.sz)/x.rel.sz)
out.df$vol<-with(out.df,e11+e22)
out.df$dva<-with(out.df,sqrt((e11-vol/2)^2+(e22-vol/2)^2))
out.df
})
ggplot(rh.eles,aes(x,y))+
geom_polygon(aes(group=id))+
coord_equal()+
theme_bw(25)
We catagorize the types of strain into two main catagories:
\[ \underbrace{\mathbf{E}}_{\textrm{Total Strain}} = \underbrace{\varepsilon_M \mathbf{I_3}}_{\textrm{Volumetric}} + \underbrace{\mathbf{E}^\prime}_{\textrm{Deviatoric}} \]
The isotropic change in size or scale of the object.
The change in the proportions of the object (similar to anisotropy) independent of the final scale
ggplot(rh.eles,aes(x,y))+ geom_polygon(aes(group=id,fill=vol),color="black")+ coord_equal()+ labs(fill="Volumetric\nStrain")+ scale_fill_gradient2()+ theme_bw(25)
ggplot(rh.eles,aes(x,y))+ geom_polygon(aes(group=id,fill=dva),color="black")+ coord_equal()+ labs(fill="Deviatoric\nStrain")+ scale_fill_gradient2()+ theme_bw(25)
This can then be expressed more concretely in the form of a strain tensor which can be broken down into volumetric and deviatoric strain \[ \mathbf{E} = \begin{bmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\ \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} \end{bmatrix} \]
__ Data provided by Goran Lovric and Rajmund Mokso __
Since lung
__ Data provided by Julie Fife __
__ Data provided by Mattia Pistone and Julie Fife __ The air phase changes from small very anisotropic bubbles to one large connected pore network. The same tools cannot be used to quantify those systems. Furthermore there are motion artifacts which are difficult to correct.
We can utilize the two point correlation function of the material to characterize the shape generically for each time step and then compare.